,log>i?ms & ,expon5tial
                ,func;ns

    ,9troduc;n & ,review4

,laws ( ,expon5ts4
  ,f/ ( all1 let's rememb] ! basic
algebraic prop]ties ( algebraic &
log>i?mic func;ns4
  ,all ! def9i;ns & !orems ab expon5ts
-e f ! funda;tal fact
  a^n"a^m .k a^n+m_4
,? pr9ciple is obvi\s if ;n & ;m >e
positive 9teg]s1 z is ! relat$ pr9ciple
t
  (a^n")^m .k a^nm_4
,"ey?+ else ab expon5tial func;ns gr[s f
a desire 6make ^! pr9ciples hold reg>d.s
( :e!r or n ;n & ;m >e positive1 or
9tegral1 or r,nal4 ,= 9/.e1 tak+ n .k #0
wd give
  a^0"a^n .k a^0+n .k a^n,
: m1ns t if we want ! expon5t (! product
6be ! sum (! expon5ts1 we h 6def9e
                                      #j
a^0 .k #1_4 ,simil>ly1 we want 6h
  a^n"a^-n .k a^0,
:1 giv5 t we n[ "k a^0 .k #1, m1ns t we
h 6def9e
  a^-n .k ?1_/a^n"#_4
  ,simil>ly1 a desire 6make
(a^n")^m .k a^nm hold univ]s,y m1ns t we
m/ h
  (a^1_/q")^q .k a^q_/q .k a^1 .k a,
: m1ns t we h 6def9e
  a^1_/q .k <q>a],
: m1ns1 9 turn1 t
  <q>a^p"] .k (a^p")^1_/q .k a^p_/q
    .k (a^1_/q")^p .k (<q>a])^p_4
  ,all ^! def9i;ns may h be5 pres5t$ 6y
9 ! pa/ j z facts 6le>n---& y %d j le>n
!m---b isn't x 9t]e/+ 6see h[ !y >ise4
,l s _m ?+s 9 ma!matics1 ! !orems came
f/---we knew :at prop]ties ( 9teg] p[]s
we want$ 6g5]alize---&! def9i;ns 7
design$ 6gu>antee ! !orems we want$4
  ,a truly obs]vant /ud5t w notice t
no?+ we h d"o tells u any?+ ab ! value (
a^r if ;r is irr,nal4 ,y've probably
                                      #a
2liev$ all yr life t expres.ns l #2^>2^]
or #10^.p make s5se1 b h y "e actu,y se5
!m def9$8 ,cd y propose def9i;ns n[8
,?9k ab ? "q2 we'll ne$ 6-e back 6x4 ,"p
(! r1son we're "w+ ) logs & expon5tial
func;ns is 6be a# 6answ] x1 : we'll d 9
a r unexpect$ way4 ,/ay tun$4

    ,log>i?ms4
  ,log>i?ms >e j expon5tial func;ns d"o
backw>ds4 ,!y h ! same rel,n 6expon5tial
func;ns t squ>e roots h 6squ>es4 ,t is1
j z
  y .k <n>x] if & only if x .k y^n,
we al h
  y .k log;a x if & only if x .k a^y_4
,! numb] log;a x is ! p[] 6: y raise ;a
9 ord] 6get ;x_4 ,?9k ? "?1 & y'll h !
two id5tities
  a^log^;a x
    .k x,
    log;a (a^x") .k x_4
  ,f ! def9i;n ( log>i?ms1 we get immly
! sort ( facts "o uses 6build 9tui;ns ab
                                      #b
logs3
  log;a #1 .k #0
  log;a a .k #1
  log;a (a^2") .k #2
  log;a (a^3") .k #3_4
  ,e law ( expon5ts is equival5t 6a law
( log>i?ms4 ,! mo/ funda;tal ( ^! -es f
! obs]v,n t
  a^log^;a (xy) .k xy
    .k (a^log^;a x")(a^log^;a y")
    .k a^log^;a x+log^;a y_4
,? m1ns t we m/ h
  log;a (xy) .k log;a x+log;a y_4
,! log (a product is ! sum (! logs (!
factors4
  ,simil>ly1 we h
  a^log^;a (x^^n^) .k x^n
    .k (a^log^;a x")^n .k a^nlog^;a x,
: m1ns t we m/ h
  log;a (x^n") .k nlog;a x_4
  ,f ^! id5tities1 we c derive o!rs (
use4 ,= 9/.e1
  log;a x+log;a ?1_/x# .k log;a ?x_/x#
    .k log;a #1 .k #0_2
                                      #c
s
  log;a ?1_/x# .k -log;a x_4
,see if y c al derive ! law
  log;a ?x_/y# .k log;a x-log;a y_4
  ,"! is a f9al id5t;y t is ess5tial 9
-put+ logs 6e35tric bases3
  log;b x .k ?log;a x_/log;a b#_4
,? lets u -pute logs 6! base ;b z l;g z
"! is "s base ;a = : we c -pute logs4
,6see t ? id5t;y is true1 we obs]ve t
  a^log^;a x .k x .k b^log^;b x
    .k (a^log^;a b")^log^;b x
    .k a^log^;a blog^;b x_4
,f ?1 we 9f] t
  log;a x .k log;a blog;b x_4
,divide bo? sides by log;a b, & y're
set4
  ,x's wor? look+ qkly at graphs (
expon5tial & log>i?mic func;ns4 ,figure
#1 %[s tgr ! graphs ( y .k #2^x &
y .k log2 ;x_4 ,s9ce y .k log2 ;x &
x .k #2^y m1n ! same ?+1 ! graph (
y .k log2 ;x &! graph ( y .k #2^x %d 2
relat$ 09t]*ang+ ! ;x & ;y axes4
                                      #d
,geometric,y1 ? 9t]*ange is a*iev$
0reflect+ ! plot acr ! l9e y .k x, : is
%[n da%$ 9 ,figure #1_4
  ,logs & expon5tial func;ns 6o!r bases
look simil> 6! graphs 9 ,figure #1_4
,figure #2 %[s ! graphs ( y .k #2^x &
y .k #8^x on ! same set ( axes4 #8^x
gr[s m* fa/] = positive ;x & %r9ks m*
fa/] = negative ;x_4 ,! correspond+
log>i?ms >e %[n 9 ,figure #3_4

    ,:y >e we do+ ?8
  ,:at has motivat$ u 6?9k n[ ab logs &
expon5tial func;ns8 ,well1 x turns \ t
we hav5't q d"o calculus z -pletely z
we've be5 claim+1 & t ^! func;ns >e a
big gap 9 \r k4 ,we've be5 claim+ =a l;g
"t1 = 9/.e1 t we c di6]5tiate any func;n
we c write d[n4 ,,ok1 if we're s cool1
:at's
  ?d_/dx#(2^x")=
,"o mi<t guess t ! answ] wd 2 x2^x-1,
?\< "o "\ 6be at l1/ a bit uneasy ab ?
answ]4 ,af all1 #2^x, ": ! v>ia# is 9 !
                                      #e
expon5t &! base is a 3/ant1 is re,y
di6]5t f x^2, ": ! v>ia# is ! base &!
expon5t is a 3/ant4 ,9 any case1 ? answ]
is hope.sly wr;g1 s9ce x wd predict t if
f(x) .k #2^x, !n f'(0) .k #0, & t
f'(x) "k #0 if x "k #0_4 ,bo? ^! claims
>e obvi\sly wr;g1 z a gl.e at ! graph (
#2^x %[s at once4
  ,s we re,y 7 premature 9 dclg victory
9 di6]5ti,n4 ,we c't di6]5tiate #2^x, &
we c't di6]5tiate log2 ;x_4 ,"!'s / "w
6be d"o4

    ,! ,natural ,log
  ,! solu;n 6! pro#m ( di6]5tiat+
expon5tial & log>i?mic func;ns -es f an
unexpect$ s\rce---\r only rema9+ 9tegr,n
pro#m 9volv+ p[]s ( ;x_4 ,rememb] t z
l;g z n /.k -#1, we h
  !x^n"dx .k ?x^n+1"_/n+1#+c_4
,:5 n .k -#1, ? =mula makes no s5se2 & s
f>1 we h n _h any m1ns = evaluat+ !
9tegral
  !?dx_/x#_4
                                      #f
  ,on ! o!r h&1 we "k t ? antid]ivative
m/ exi/1 s9ce 0! ,funda;tal ,!orem (
,calculus1 "o antid]ivative ( ?1_/x# is
! >ea func;n
  !;1^x"?dt_/t#_4
,\r goal at ! mo;t is 6see :at we c le>n
ab ? >ea func;n4
  ,if y plot ! >ea "u ! graph ( ?1_/t#
2t t .k #1 & t .k x, y get ! graph %[n 9
,figure #4, : looks a lot l a log>i?m4
  ,! amaz+ ?+ is t ? >ea func;n is a
log>i?m6 ,i f9d ? "o (! mo/ />tl+
results 9 ele;t>y calculus4 ,all !
antid]ivatives ( p[]s ( ;x >e j simple &
un9t]e/+ multiples ( o!r p[]s ( ;x_4
,!n1 -pletely \ (! blue1 "! is "o
excep;n3 ! antid]ivative ( x^-1, al"o \
( all ! 9f9itely _m p[]s ( x, turns \
6be a -pletely di6]5t k9d ( func;n1 a
log>i?m4 ,? "o 9tegral lives 9 an utt]ly
di6]5t univ]se f all ! o!rs1 & op5s up
:ole new ma!matical r1lms4 ,we've r
casu,y ignor$ ? "o excep;nal 9tegral up
until n[1 b "o ignores s* o4balls at
                                      #g
"o's p]il4 ,li/5+ 6! message ( ? "o
func;n \ ( 9f9itely _m t m>*es 6! b1t (a
di6]5t drumm] op5s new _ws 6\r view4
  ,,ok1 s let's def9e ! natural log
func;n z
  ln x .k !;1^x"?dt_/t#,
! >ea "u ! graph ( y .k ?1_/t# 2t
t .k #1 & t .k x_4 ,! graph ( ? func;n
is at l1/ r\<ly l ! graph ( y .k log;a x
= "s ;a_4 ,we n[ ne$ 6%[ t 9 fact1 ln x
is a log>i?m func;n4
  ,let's />t \ 0%[+ t ln x has ! "r
algebraic prop]ties 6be a log>i?m 6"s
base4 ,"o ( ^! is alr obvi\s3
  ln #1 .k !;1^1"?dt_/t# .k #0_4
,m sub/antively1 ?\<1 we'd l 6%[ ?+s l
ln (ab) .k ln a+ln b_4 ,6d ?1 we c />t \
0notic+ t
  ln (ab) .k !;1^ab"?dt_/t#
    .k !;1^a"?dt_/t#+!;a^ab"?dt_/t#
    .k ln a+!;a^ab"?dt_/t#_4
,we'd "!=e 2 d"o if we cd %[ t
  !;a^ab"?dt_/t# .k ln b_4
,? turns \1 ?\<1 j 6be an easy 9tegr,n
                                      #h
0sub/itu;n3 ! sub/itu;n
  u .k ?t_/a#
  du .k ?dt_/a#,
or1 6put x m 3v5i5tly1
  t .k au
  dt .k adu
turns ! 9tegral 9to
  !;a^ab"?dt_/t# .k !;1^b"?adu_/au#
    .k !;1^b"?du_/u# .k ln b,
s
  ln (ab) .k ln a+!;a^ab"?dt_/t#
    .k ln a+ln b_4
  ,simil>ly1 we'd l 6"k t
ln ?1_/a# .k -ln a_4 ,? ag is a 3sequ;e
( sub/itu;n4 ,! sub/itu;n
  u .k at
  du .k adt,
or1 6put x m 3v5i5tly1
  t .k ?u_/a#
  dt .k ?du_/a#
lets u write
  ln ?1_/a# .k !;1^1_/a"?dt_/t#
    .k -!;1_/a^1"?dt_/t#
    .k -!;1^a",?du,_/a(u_/a),#
                                      #i
    .k -!;1^a"?du_/u# .k -ln a_4
  ,! id5t;y ln (a^n") .k nln a is al a
3sequ;e (a sub/itu;n l ?4 ,if we use !
sub/itu;n
  u .k t^1_/n
  du .k ?t^?1_/n#-1"_/n#dt,
or1 m simply1
  t .k u^n
  dt .k nu^n-1"du,
!n we f9d
  ln (a^n") .k !;1^a^^n"?dt_/t#
    .k !;1^n"?nu^n-1"du_/u^n"#
    .k n!;1^a"?du_/u# .k nln a_4

  ,all ? makes x seem hi<ly likely t !
func;n ln x re,y is a log 6"s base4 ,b
:at is ! base (! natural log>i?ms8
,well1 if ln x is re,y log;e x = "s
numb] ;e, !n x m/ 2 t
ln e .k log;e e .k #1_4 ,9 o!r ^ws1 ;e
m/ 2 ! numb] = :
  #1 .k ln e .k !;1^e"?dt_/t#_4
  ,6approximate ;e, we cd 2g9 0j us+
rectangles 6approximate ! 9tegral4 ,=
                                     #aj
9/.e1 if we use left rectangles ( wid?
?1_/4#, we f9d t
  !;1^2.25"?dt_/t#
    "k ?1_/4#.(?1_/1#+,?1,_/5_/4,#
    +,?1,_/3_/2,#+,?1,_/7_/4,#+?1_/2#
    +,?1,_/9_/4,#.) "k #1,
s t e .1 #2.25_2 & if we use "r
rectangles ( wid? ?1_/4#, we f9d t
  !;1^3"?dt_/t#
    .1 ?1_/4#.(,?1,_/5_/4,#+,?1,_/3_/2,#
    +,?1,_/7_/4,#+?1_/2#+,?1,_/9_/4,#
    +,?1,_/5_/2,#+,?1,_/11_/4,#
    +?1_/3#.) .1 #1
s t e "k #3_4 ,bett] calcul,ns ) m
rectangles or ) o!r expres.ns = ;e give
  e ".k<*] #2.718281828459_4
  ,if ?p_/q# is any r,nal numb]1 !n !
id5tities we've j prov$ =! natural log
func;n give
  ln (e^p_/q") .k (?p_/q#)ln e
    .k (?p_/q#)1 .k ?p_/q#_4
,?us1 ! natural log func;n seems 6be !
log 6! base ;e, = e numb] x .k ?p_/q# at
: e^x is def9$4
                                     #aa
ln (x) & 9tegrals4
  ,we n[ h a func;n ln x def9$ = x .1 #0
& hav+ ! prop]ty t :5 x .1 #0,
?d_/dx#ln x .k ?1_/x#, i4e41 t :5
x .1 #0,
  !?1_/x#dx .k ln x+c_4
,:at c we say ab ! 9tegral ( ?1_/x# if
x "k #0_8 ,well1 if x "k #0, !n
-x .1 #0_2 s 0! ,*a9 ,rule1
  ?d_/dx#ln (-x) .k ?1_/-x#(-1)
    .k ?1_/x#_4
,? m1ns t = negative ;x,
  !?1_/x#dx .k ln (-x)+c_4
,we c roll ^! two obs]v,ns tgr 6say t =
any x /.k #0,
  !?1_/x#dx .k ln \x\+c_4
,?us1 \r def9ite 9tegral =! func;n
?1_/x# />t+ at x .k #1 al yields an
9def9ite 9tegral = ?1_/x# valid at e
x /.k #0_4
  ,>m$ ) ?1 we c d lots ( o!r 9tegrals z
well4 ,= 9/.e1 3sid]
  !?2x-3_/x^2"-3x+1#dx_4
,? 9tegral c 2 reduc$ 0! sub/itu;n
                                     #ab
  u .k x^2"-3x+1
  du .k (2x-3)dx
6give
  !?du_/u# .k ln \u\+c
    .k ln \x^2"-3x+1\+c_4
,ano!r cl"e example (! same te*nique
lets u 9tegrate
  !tan xdx_4
,we rewrite ? 9tegral z
  !?sin x_/cos x#dx,
!n use ! sub/itu;n
  u .k cos x
  du .k -sin xdx
6write ? z
  -!?du_/u# .k -ln \u\+c
    .k -ln \cos x\+c_4
,! f/ ?+ y wd h guess$1 "r8

,us+ ln x 6def9e expon5ti,n4
  ,:5 we talk$ 9=m,y ab expon5tial &
log>i?mic func;ns1 we describ$ !m z
9v]ses ( "o ano!r1 j z ! squ>e root &
squ>e func;ns >e 9v]ses ( "o ano!r4 ,if
! func;ns e^x & ln x 7 6be 9v]ses ( "o
                                     #ac
ano!r1 :at wd we ne$8 ,funda;t,y1 all we
wd require algebraic,y wd 2 t ^! two
func;ns undo "o ano!r1 t is1 t
  ln (e^x") .k x
& t
  e^ln x .k x_4
  ,d we "k ^! facts8 ,well1 we "k t if
x .k ?p_/q# is r,nal1 !n
  ln (e^x") .k ln (e^p_/q")
    .k ?p_/q#ln (e) .k ?p_/q##1
    .k ?p_/q# .k x_4
,we don't "k t ? same =mula holds if ;x
is "s irr,nal numb]1 l >2] or l .p, b !
r1son we don't "k x is "s:at surpris+4
,we don't "k x's true 2c we don't "k :at
expres.ns l e^>2^] & e^.p m1n6 ,we h n"e
def9$ ^! expres.ns 9 s*ool2 we've alw j
2hav$ z if !y probably made s5se4
  ,! natural log f9,y gives u a *.e
6rectify ? ov]si<t4 ,suppose we def9e !
func;n exp (x) 6be ! 9v]se func;n (
ln (x)_4 ,t is1 we say1 let exp (x) 2 !
func;n def9$ s t
  y .k exp x if & only if x .k ln y_4
                                     #ad
,! func;ns ln  & exp  undo "o ano!r3
  ln (exp (x)) .k x & exp (ln (x))
    .k x_4
,l ! graphs ( any pair ( 9v]se func;ns1
! graphs ( y .k ln x & y .k exp (x) >e
relat$ 0reflec;n acr ! l9e y .k x_4 ,all
( ? makes s5se 2c ln x is an 9cr1s+
func;n1 s t e horizontal l9e crosses xs
graph at mo/ once4 (,? is s 2c xs
derivative1 ?d_/dx#ln x .k ?1_/x# .1 #0
= all x .1 #0_4)
  ,:at c we n[ say ab ! func;n
y .k exp (x)_8 ,well1 if x .k ?p_/q# is
r,nal1 !n we "k t
  ln (exp (?p_/q#)) .k ?p_/q#
    .k ln (e^p_/q")_4
,s9ce ln  is an 9cr1s+ func;n1 ? c only
happ5 if
  exp (?p_/q#) .k e^p_/q_4
,we al "k t ln x is a 3t9u\s func;n
def9$ at e x .1 #0, & t xs range is !
set ( all r1l numb]s4 ,? m1ns t xs 9v]se
func;n exp (x) m/ al 2 3t9u\s1 & t
exp (x) is def9$ at e r1l numb] ;x_4 ,s
                                     #ae
exp (x) is a 3t9u\s func;n def9$ at e ;x
& equal to e^x at e po9t at : e^x is
def9$4 ,s :y n d ! natural ?+ "h8 ,:y n
def9e
  e^x .k exp (x)
= e r1l numb] ;x_8 ,we hav5't *ang$ !
value ( e^x any":1 b we've n[ giv5 e^x a
value at all ^? irr,nal po9ts ": x
curr5tly lacks "o4 ,? is gd4

,! derivative (! expon5tial func;n4
  ,"! >e n[ two approa*es 6-put+ !
derivative (! func;n y .k e^x_4 ,! f/ is
6be algebraic,y cl"e1 & 6r1son l ?3 ,we
rememb] t ln x is an antid]ivative (
?1_/x#, s t
  ?d_/dx#ln x .k ?1_/x#_4
,we al "k t
  ln (e^x") .k x_4
,tak+ ! derivative ( bo? sides us+ ! *a9
rule gives
  ?1_/e^x"#*?d_/dx#(e^x") .k #1
  ?d_/dx#(e^x") .k e^x_4
,a pretty wild result3 e^x is xs [n
                                     #af
derivative6 ,? makes s5se if y ?9k ab !
graph ( y .k e^x, %[n 9 ,figure #5_4 ,!
slope (! tang5t l9e 6? func;n is close
6z]o = negative ;x & gr[s rapidly =
positive ;x, ! same z ! func;n xf4
  ,a second approa* 6-put+ ! derivative
(! expon5tial func;n is 63sid] tgr !
graphs ( y .k e^x & y .k ln x, : >e %[n
9 ,figure #6_4
  ,! po9t (x, e^x") on ! graph (
y .k e^x c 2 reflect$ acr ! l9e y .k x
6yield a po9t (e^x, x) on ! graph (
y .k ln x_4 ,! tang5t l9e 6! graph (
y .k ln x at ? po9t has slope
  y'(e^x") .k ?1_/e^x"#_4
,x is al cle> geometric,y t ! slope (!
tang5t l9e to y .k e^x at ! po9t (x, e^x
") is ! reciprocal (! slope (! tang5t
l9e to y .k ln x at ! po9t (e^x, x)_4 ,9
algebra1
  ?d_/dx#(e^x") .k ,?1,_/1_/e^x",#
    .k e^x,
! same result we got 2f4

                                     #ag
,a celebr,n4
  ,? /uff has all be5 te*nical & subtle1
b we've made a fanta/ic am.t ( progress
9 a few pages4 ,we've manag$ 6-pute an
important 9tegral we cdn't -pute 2f4
,we've def9$ at l1/ e^x = all ! 9f9itely
_m irr,nal numb]s---9 fact1 ! ov]:elm+
major;y ( all r1l numb]s---= : x 0
previ\sly undef9$4 ,we've su3ess;lly
di6]5tiat$ bo? ln x & e^x_4 ,& we've f.d
a func;n t's xs [n derivative---! sort (
?+ "! "\ 6be a use =4
  ,let's celebrate4

    ,o!r ,bases = ,logs & ,expon5tial
    ,func;ns
  ,we've gott5 all te>y ey$ ab 2+ a#
6def9e & d calculus on ! func;ns e^x &
ln x, b :at ab o!r expon5tial & log
func;ns8 ,af all1 we didn't c>e ab (or
ev5 "k ab) e^x & ln x until rec5tly4
,:at ab #10^x & log10 ;x, or #2^x &
log2 ;x_8

                                     #ah
  ,! answ]s turn \ 6be q simple4 ,z l;g
z x .k ?p_/q# is r,nal1 s t "ey?+ is
def9$1 we c write
  #2^x .k (e^ln #2")^x .k e^xln #2_4
,:5 ;x is irr,nal1 we don't yet h a
def9i;n = #2^x_4 ,s :y n d ! natural ?+
& def9e
  #2^x .k e^xln #2
= all ;x_8 ,we won't h *ang$ ! def9i;n (
#2^x at any po9t ": x is curr5tly def9$1
& we'll h 5d$ up )a 3t9u\s func;n we "k
h[ 6di6]5tiate4 ,af all1 ! *a9 rule
tells u t
  ?d_/dx#(2^x")
    .k ?d_/dx#(e^xln #2") .k ?d_/dx#(
      exp (xln #2)) .k exp (xln #2)*?d
      _/dx#(xln #2)
    .k exp (xln #2)*ln #2 .k e^xln #2"
      ln #2 .k #2^x"ln #2_4
,! same ?+ is true = e o!r positive
3/ant ;a_4
  ?d_/dx#(a^x") .k a^x"ln a_4
  ,we c d "s?+ simil> ) log>i?ms4 ,!
3di;n t y .k #2^x %d 2 equival5t to
                                     #ai
x .k log2 ;y_4 ,b x is al equival5t to
y .k e^xln #2, : is equival5t to
ln y .k xln #2, or to
x .k ?ln y_/ln #2#_4 ,x wd "!=e 2 s5si#
6def9e
  log2 y .k ?ln y_/ln #2#,
: wd m1n t
  ?d_/dy#(log2 ;y)
    .k ?d_/dy#(?ln y_/ln #2#)
    .k ?1_/yln #2#_4
,! same ?+ is true ( any o!r positive
base1 ;a_4
  ?d_/dy#(log;a y)
    .k ?d_/dy#(?ln y_/ln a#)
    .k ?1_/yln a#_4
  ,n[ we c def9e & di6]5tiate e
expon5tial or log>i?mic func;n "ey":4

    ,:at's ,natural ,ab ,natural ,logs8
  ,:5 "o f/ says 6p1 8,! natural log is
! log base #2.718281828459 ''' ,_0 !
usual reac;n is "o ( quiet amuse;t4 8
,wdn't1 say1 #10, 2 ev5 m natural80 "o c
he> !m ?9k+4 ,if "o ?9ks ab x1 ?\<1 "o c
                                     #bj
0n[ >ticulate lots ( r1sons :y ln x &
e^x >e func;ns t ma!maticians wd alm h
6h 9v5t$ & /udi$1 :ile ! "picul> *oice (
#10^x & log10 ;x reflects :at wd seem
6be 3t+5t evolu;n>y *oices made 0\r
amphibian ance/ors4
  ,"o way 9 : ;e is ! simple/ base =
log>i?ms & p[]s is se5 9 ! derivative
=mulas
  ?d_/dx#ln x .k ?1_/x#
  ?d_/dx#e^x .k e^x,
: >e simpl] ?an ! analog\s =mulas = any
o!r base1
  ?d_/dx#log;a x .k ?1_/xln a#
  ?d_/dx#a^x .k a^x"ln a_4
,us+ any o!r base ?an ;e 5tails c>ry+
>.d a bun* ( factors ( an annoy+ 3/ant1
ln a, : >5't pres5t if y use base ;e_4
,s base ;e looks simpl] & m natural4 ,n
only t1 b ! annoy+ 3/ant ln a y're c>ry+
>.d is def9$ 9 t]ms (! natural log1 :
m1ns t ev5 if y didn't />t \ 9t]e/$ 9
natural logs1 y'd soon 2 =c$ 69v5t !m 9
ord] 6"u/& h[ t 3/ant dep5d$ on ;a_4 ,s
                                     #ba
x looks l y c't d1l ) derivatives ( logs
or p[]s at all )\t ev5tu,y 9v5t+ !
natural log4
  ,"o way ( say+ ? simply 0?9k+ ab
expon5tial func;ns is 6say t = any a, !
func;n a^x is a func;n propor;nal 6xs [n
derivative4 ,3front$ )a :ole raft ( ^!
func;ns ) di6]5t 3/ants ( propor;nal;y1
wdn't "o natur,y look =! "o ": ! 3/ant (
propor;nal;y 0 #1, i4e41 ! func;n t 0
equal 6xs [n derivative8 ,? func;n t x
wd 2 natural 6/udy is e^x_4
  ,( c\rse1 "o cd al >gue t af f9d+
antid]ivatives = e p[] ( ;x except =
x^-1, x wd 2 pretty natural 6look = an
antid]ivative = x^-1, : wd probably 5d
up ) u 9v5t+ ln x = ? purpose al"o4
  ,9 %ort1 ev5 ?\< "! is no way 6claim t
;e is ! f/ numb] a naive p]son wd pick \
z a base = log>i?ms1 ! important
func;nal prop]ties ( ln x & e^x wd seem
6make _! 9v5;n 9evita# 9 any society
develop+ calculus4 ,no o!r base has ?
prop]ty1 s 9 a v r1l s5se1 ln x is .!
                                     #bb
natural log>i?m4

    ,a ,glimpse ( ,di6]5tial ,equ,ns &
    ,expon5tial ,func;ns4
  ,! fact t ! derivative ( e^x is e^x or
t ! derivative ( a^x is a^x"ln a, a
3/ant multiple ( a^x, makes expon5tial
func;ns v important 9 a :ole range (
applic,ns4 ,if ;y is a func;n ( "t
repres5t+ ! popul,n (a species dur+ gd
"ts1 or ! am.t ( m"oy 9 a sav+s a3.t 9
gd "ts1 or ! am.t (a radioactive mat]ial
t has n yet decay$1 !n "o ?+ we "k ab ;y
is t xs derivative is propor;nal 6xf3
  y' .k ry_4
,! 3/ant ;r is ! gr[? rate (! popul,n1
or ! 9t]e/ rate (! a3.t1 or ! decay
3/ant (! radioactive mat]ial4
  ,"! >e fanci] ways 6>rive at ? 3clu.n1
b "o way 6solve ? di6]5tial equ,n is j
6guess4 ,! func;n y .k e^rt solves
y' .k ry_4 ,s does
  y .k ,ce^rt
= e 3/ant ;,c_4 ,)a small am.t ( "w1 "o
                                     #bc
c %[ t ^! >e ! only solu;ns 6? ,,de4
,rememb] t y saw x "h f/4

    ,a f9al bit ( amuse;t3 x^x_4
  ,let ;a 2 a 3/ant4 ,=a l;g "t1 we've
"kn h[ 6di6]5tiate x^a_4
  ?d_/dx#(x^a") .k ax^a-1_4
,we've j n[ le>n$ z well h[ 6di6]5tiate
a^x_4
  ?d_/dx#(a^x") .k a^x"ln a_4
  ,,ok1 s :at ab x^x_8 ,:at's xs
derivative8

  #2 .wr;g .answ]s '''

  ,maybe x^x is l x^a, 9 : case
?d_/dx#(x^x") .k xx^x-1 .k x^x_4
  ,or maybe x^x is l a^x, 9 : case
?d_/dx#(x^x") .k x^x"ln x_4

  ..''' &a "r ."o4

  ,0! def9i;n we've giv5 =! expon5tial
func;n1
                                     #bd
  x^x .k e^xln x .k exp (xln x)_4
,xs derivative is "!=e
  ?d_/dx#(e^xln x")
    .k (e^xln x")(?d_/dx#(xln x))
    .k (e^xln x")(ln x+x?1_/x#)
    .k (e^xln x")(1+ln x)
    .k x^x"(1+ln x)
    .k x^x"+x^x"ln x_4
,! "r answ] is ! sum (! two wr;g "os4
,x's a p;y ? doesn't "w m g5],y6














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